fluid_mechanics

6.1 Fluid Element Kinematics 265
x
^i
z
v
^k
^j
V
w
The acceleration of
a fluid particle is
described using the
concept of the material
derivative.
u
y
6.1.1 Velocity and Acceleration Fields Revisited
As discussed in detail in Section 4.1, the velocity field can be described by specifying the velocity
V at all points, and at all times, within the flow field of interest. Thus, in terms of rectangular
coordinates, the notation V 1x, y, z, t2 means that the velocity of a fluid particle depends on where
it is located within the flow field 1as determined by its coordinates, x, y, and z2 and when it
occupies the particular point 1as determined by the time, t2. As is pointed out in Section 4.1.1,
this method of describing the fluid motion is called the Eulerian method. It is also convenient to
express the velocity in terms of three rectangular components so that
V uî vĵ wkˆ
(6.1)
where u, and w are the velocity components in the x, y, and z directions, respectively, and
î, ĵ, and kˆ v, are the corresponding unit vectors, as shown by the figure in the margin. Of course,
each of these components will, in general, be a function of x, y, z, and t. One of the goals of
differential analysis is to determine how these velocity components specifically depend on x, y,
z, and t for a particular problem.
With this description of the velocity field it was also shown in Section 4.2.1 that the
acceleration of a fluid particle can be expressed as
and in component form:
The acceleration is also concisely expressed as
where the operator
D1 2
Dt
a 0V
0t
01 2
0t
is termed the material derivative, or substantial derivative. In vector notation
where the gradient operator, 1 2, is
D1 2
Dt
u 0V
0x v 0V
0y w 0V
0z
a x 0u
0t u 0u
0x v 0u
0y w 0u
0z
a y 0v
0t u 0v
0x v 0v
0y w 0v
0z
a z 0w
0t u 0w
0x v 0w
0y w 0w
0z
a DV
Dt
u 01 2
0x v 01 2
0y w 01 2
0z
01 2
0t
1V 21 2
1 2 01 2
0x î 01 2
0y ĵ 01 2
0z kˆ
(6.2)
(6.3a)
(6.3b)
(6.3c)
which was introduced in Chapter 2. As we will see in the following sections, the motion and
deformation of a fluid element depend on the velocity field. The relationship between the motion
and the forces causing the motion depends on the acceleration field.
(6.4)
(6.5)
(6.6)
(6.7)
6.1.2 Linear Motion and Deformation
The simplest type of motion that a fluid element can undergo is translation, as illustrated in Fig.
6.2. In a small time interval dt a particle located at point O will move to point O¿ as is illustrated
in the figure. If all points in the element have the same velocity 1which is only true if there are no
velocity gradients2, then the element will simply translate from one position to another. However,